Non-existence of time-periodicvacuum spacetimes

Volker Schlue

(joint work with Spyros Alexakis and Arick Shao)

Universite Pierre et Marie Curie (Paris 6)

Dynamics of self-gravitating matter workshop,IHP, Paris, October 26-29, 2015

Outline

1 2-body problem in general relativityPredictions of post-Newtonian theory

2 Time-periodic vacuum spacetimesStatement of the main theorem

3 Final state conjectureDynamical non-radiating spacetimesRelation to black hole uniqueness problem

4 Proof of the non-existence resultUniqueness results for ill-posed hyperbolic p.d.e.’sNull geodesics in spacetimes with positive mass

1 2-body problem in general relativity

Periodic motion in Newton’s theory

Two-body problem in classical celestial mechanics: Kepler orbits.Space: R3, Time: R, Newtonian potential: ψ(t, x)

F = −m∇ψ(t, x)

m

ρ(t, x)

Sun

Earth

Test body:

4ψ(t, x) = 4πρ(t, x) : mass density

Note: In Newtonian theory the gravitational field can be periodicin time. (Action at a distance)

Post-Newtonian theory

Slow motion / post-Newtonian / weak field approximations:

Einstein-Infeld-Hoffmann, . . . , Damour-Deruelle,

Blanchet, . . . , Will-Wiseman, Poisson-Will

Sun

Earth

Correction to Newtonian potential: ψpost-Newtonian = ψ + 1c2ω

In this approximation circular orbits still possible, but ruled out by

radiation reaction force in higher orders of the expansion.

Dissipative dynamicsin general relativity

Periodic motion should not exist in general relativity due to theemission of gravitational waves.

Our main result (joint with Spyros Alexakis) is thatany time-periodic vacuum spacetime is in fact time-independent,at least far away from the sources.

2 Time-periodic vacuum spacetimes

Isolated self-gravitating systemsin general relativity

(M3+1, g) asymptotically flat, and solution to Ric(g) = 0 outsidea spatially compact set.

Ric(g)=0

I+

I−

ι0

source

: future null infinity

: spacelike infinity

Future and past null infinities I+, I− ' R× S2 are complete.

Notion of time-periodicity

Definition

An asymptotically flat spacetime is called time-periodic if thereexists a discrete isometry ϕa with time-like orbits.

p

ϕa(p)

I−

ι0

I+ ' R× S2

ϕ+a (u, ξ) = (u+ a, ξ)

ϕa

Then in fact ϕa extends to a map ϕ+a on future null infinity where

it is an affine translation along the generating geodesics. (Similarlyat past null infinity.)

Theorem ( Alexakis-S. ’15)

Any asymptotically flat solution (M, g) to the Einstein vacuumequations arising from a regular initial data set which istime-periodic (near infinity), must be stationary near infinity.

I+

I−

ι0 D T ι0

The theorem asserts that there exists a time-like vectorfield T on an

arbitrarily small neighborhood D of infinity such that LTg = 0 on D.

Previous results

Early work:

Papapetrou ’57-’58 (weak field approximation, “non-singular”solutions, strong time-periodicity assumption)

Recent work:Gibbons-Stewart ’84, Bicak-Scholz-Tod ’10 (containsideas how to exploit time-periodicity, stationarity inferred undermuch more restrictive analyticity assumption)

Cosmological setting:

Tipler ’79, Galloway ’84 (spatially closed case)

3 Dynamical non-radiating spacetimesand the final state conjecture

Gravitational radiation

I+

I−

ι0Σ

Cu

Trautmann-Bondi energy:

The Bondi mass M(u) signifies the amount ofenergy in the system at time u. It is known tobe positive: M(u) ≥ 0 (Schoen-Yau,. . . ,

Chrusciel-Jezierksi-Leski, Sakovich),and dynamically monotone decreasing.

In Christodoulou-Klainerman the Bondi mass loss formula is

∂M(u)

∂u= − 1

32π

∫S2

|Ξ|2dµγ

and it is shown that limu→−∞M(u) = M[Σ] limu→∞M(u) = 0.

|Ξ|(u, ξ): power of gravitational waves radiated in directionξ ∈ S2, at time u ∈ R.

Aside: Gravitational wave experiments

Figure: LIGO, Washington

v

v vB(A) =d0

rΞAB(t)

Non-radiating spacetimes

Definition

An asymptotically flat spacetime is called non-radiating if theBondi mass M(u) is constant along future (and past) null infinity.

Theorem ( Alexakis-S. ’15)

Any asymptotically flat solution (M3+1, g) to the vacuumequations arising from regular initial data which is assumed to benon-radiating, and in addition smooth at null infinity,must be stationary near infinity.

Remark:Here smooth at null infinity means in particular that the curvaturecomponents ρ admit a full asymptotic expansion near null infinitywhich is well behaved towards spacelike infinity:

ρ ∼∞∑l=0

κl(u)rk−l limu→−∞

|κl(u)| <∞ .

Conjectures

The final state conjecture gives a characterisation of all possibleend states of the dynamical evolution in general relativity,as a result of a scenario due to Penrose invoking both weakcosmic censorship and black hole uniqueness.

Conjecture

Any smooth asymptotically flat black hole exterior solution to theEinstein vacuum equations which is assumed to be non-radiatingis isometric to the exterior of a Kerr solution (M, gM,a).

I+H+

Σ

D

I+: future null infinity

H+: future event horizon

D: black hole exterior

Aside: Soliton resolution conjecture

The energy-critical focusing non-linear wave equation

φ = −φ5

has soliton solutions

φλ(t, x) =1

λ12

W(xλ

)W (x) =

(1 +|x |2

3

)− 12

λ > 0

It is expected that for global solutions

‖φ−(φL ±

∑j

φλj

)‖ −→ 0 (t →∞)

Established for radial solutions by Duyckaerts-Kenig-Merle ’13.

4 Proof of the theorems:extension of a time translation symmetry from infinity

Strategy of Proof

1 Construction of time-like candidate vectorfield T , such thatby time-periodicity

limr→∞

rkLTR = 0 ∀k ∈ N

2 Use that by virtue of the vacuum Einstein equations,

gR = R ∗ R

thus“ gLTR = R ∗ LTR ” .

Then use our unique continuation theorem which assertsthat solutions to wave equations on asymptotically flatspacetimes are uniquely determined if all higher orderradiation fields are known, to show that

LTR = 0 .

Construction of “candidate” Killingvectorfield T

Σ0

Bd∗ S0

C−∗

C+0

S∗0

C+u

LL

T

Su,sSu,s

Define T = ∂∂u : binormal to spheres S∗u . Then extend inwards by

Lie transport along geodesics: [L,T ] = 0, ∇LL = 0, g(L, L) = 0.

Time-periodicityand time-independence to leading order

The components of the curvature fall off at different rates in thedistance (peeling):

RLL = O(r−1) RLL = O(r−2) RLL = O(r−3)

limu;r→∞

rRLL = A limu;r→∞

r2RLL = P limu;r→∞

r3RLL = A

Using the asymptotic laws obtained inChristodoulou-Klainerman it follows

Ξ = 0 =⇒ A = −∂uΞ = 0, ∂uP = −A = 0

however, in general,∂uA = ∇ξP + A .

Time-periodicityand time-independence to leading order (continued)

The idea is to differentiate a second time,

∂2uA = ∇ξ∂uP = 0

which implies that A is a linear function in u,

A(u2, ξ)− A(u1, ξ) = A0(ξ)(u2 − u1) .

But by time-periodicity A0(ξ) = 0, therefore

∂uA = 0 .

Note, same conclusion if instead of time-periodicity we assume

limu1→−∞

|A(u1, ξ)| <∞

This also yields ∇ξP = 0, namely that P is spherically symmetric.

Time-periodicityand time-independence to all orders

Schematically, this was the first step of an induction which proves

limu;r→∞

rk∂uR = 0 ∀k ∈ N .

In fact, we show at the same time using the propagationequations along outgoing null geodesics

limu;r→∞

rk∂ug = 0 limu;r→∞

rk∂uΓ = 0

For example, consider Γ = χ. (Recall we already saw Ξ = lim r χ.)Schematically, since by construction [L,T ] = 0, T = ∂u:

Lχ = −α α = RLL

L∂uχ = −∂uα

Time-periodicityand required regularity

In the time-periodic setting therequired regularity can bededuced from a correspondingregularity assumption on theinitial data:

g |Σ = gKerr|t=0 + g∞

g∞αβ ∼∑l

g lαβ(ϑ)rk−l

D

ι0

I+

I−

ΣΩ

Cu

Recall strategy of the proof

1 We have now constructed a time-like candidate vectorfield T ,such that by time-periodicity, or alternatively by assuming aregular expansion,

limr→∞

rkLTR = 0 ∀k ∈ N

2 Use that by virtue of the vacuum Einstein equations

gR = R ∗ R

the Lie derivative of the curvature satisfies a wave equation

“ gLTR = R ∗ LTR ” .

Then apply our unique continuation theorem to show that

LTR = 0 .

Unique continuation from infinity

Theorem ( Alexakis-S.-Shao ’14)

Let (M, g) be an asymptotically flat spacetime with positivemass, and Lg a linear wave operator

Lg = g + a · ∇+ V

with suitably fast decaying coefficients a, and V . If φ is a solutionto Lgφ = 0 which in addition satisfies∫

Drkφ2 + rk |∂φ|2 <∞

where D is an arbitrarily small neighborhood of infinity ι0, then

φ ≡ 0 : on D′ ⊂ D .

Application of the theoremto the Einstein equations

The application of the theorem is not immediate because

gR = R ∗ R

is not a scalar equation, but a covariant equation for theRiemann curvature tensor. Moreover, [g ,LT ] 6= 0 anddifferentiating the equation produces additional terms which arenot in the scope of the theorem:

gLTR − [g ,LT ]R = R ∗ LTR + LTg ∗ R2

These obstacles can be overcome in the general framework ofIonescu-Klainerman ’13 for the extension of Killingvectorfields in Ricci-flat manifolds.

Application of the theoremIonescu-Klainerman approachDefine modified Lie-derivative

W = LTR − B · R B = LTg + ω ∇Lω =M(LTg)

which then satisfies a covariant equation

W = R ·W +∇R · ∇B + R2 · B + R · ∇P

coupled to o.d.e.’s

∇LB =M(P,B) ∇LP =M(W ,B,P)

This is only true if [L,T ] = 0, which we have by construction.

Now choose Cartesian coordinates near infinity, such that

Γ = O(r−2) .

and apply our Carleman estimates.

4 Proof of the theorems:unique continuation from infinity for linear waves

Linear theory on Minkowski spaceConsider the linear wave equation on R3+1:

φ = 0

The radiation field is defined by

Ψ(u, ξ) = limr→∞

(rφ)(t = u + r , x = rξ) .

Theorem ( Friedlander ’61)

For solution φ = 0 there is a 1-1 correspondence

(φ|t=0, ∂tφ|t=0) ∈ H1 × L2 ←→ Ψ ∈ L2 .

However, no generalisations to perturbations of Minkowski spaceare known, i.e. for equations

gφ+ a · ∇φ+ Vφ = 0

Counterexamples in linear theory

Question:

Without the finite energy condition, does the vanishing of theradiation field imply the vanishing of the solution?

Ψ = 0 =⇒ φ ≡ 0

Answer:No, because φ = 1

r is a solution, and thus also

φi = ∂x i1

r∼ 1

r2

which is a non-trivial solution with Ψ = 0.

This shows the necessity of the infinite order vanishing assumption:

limr→∞

(rkφ)(u + r , rξ) = 0 ∀k ∈ N .

Obstructions to unique continuationfrom infinity

There is an obstruction related to the behavior of light rays.

Alinhac-Baouendi ’83: Unique continuation fails acrosssurfaces which are not pseudo-convex, in general.

Unique continuation from infinity forlinear waves

Theorem ( Alexakis-S.-Shao ’13)

Let (M, g) be a perturbation ofMinkowski space, and Lg a linear waveoperator with decaying coefficients. If φis a solution to Lgφ = 0 which inaddition satisfies an infinite ordervanishing condition on “at least half” offuture and past null infinity, then

φ ≡ 0

in a neighborhood of infinity.

D

I−ε

I+ε

Pseudo-convexity

The proof crucially relies on the construction of a family ofpseudo-convex time-like hypersurfaces.

Definition

A time-like hypersurface H = f = c ispseudo-convex at a point p, if

(∇2f )p(X ,X ) < 0

for all vectors X ∈ TpM which(i) are null, g(X ,X ) = 0,(ii) are tangential to H, g(X ,∇f ) = 0.

p

We find a family of pseudo-convex hypersurfaces that foliate aneighborhood of infinity and derive a Carleman inequality toprove the uniqueness result.

Conformal Inversionin Minkowski space

In fact, we choose

f =1

(−u + ε)(v + ε)u =

1

2

(t − r

), v =

1

2

(t + r

)and consider the conformally inverted metric

g = f 2g

ι0

I+

I−

INote this is not thestandard Penrosecompactification Ω2gwhere

Ω =1√

(1 + u2)(1 + v2)

In fact g is singular.

Conformal Inversionin Schwarzschild

In Schwarzschild with m > 0,

g = −4(

1− 2m

r

)dudv + r2γ

where, for arbitrary r0 > 2m,

v − u = r∗ =

∫ r

r0

1

1− 2mr

dr = r + 2m log|r − 2m| − r∗0

t = 0

u = 0

v = 0

r = r0

ι0

I+r0

I−r0

We set

f =1

(−u)(v)

and consider theconformally invertedmetric

g = f 2g .

Pseudo-convexityin spacetimes with positive mass

While in Minkowski space

−∇2f (X ,X ) ∼ ε

r∀X : g(X ,X ) = g(X ,∇f ) = 0

we find that in Schwarzschild

−∇2f (X ,X ) ∼ 2m

rlog r − r0

r> 0

for arbitrarily large r0 > 2m.

This is the reason unique continuation from infinity holds in anarbitrarily small neighborhood of infinity whenever thespacetime has a positive mass.

Positive massand the behaviour of light rays

ι0M > 0

light ray

This is related to ideas of Penrose, Ashtekar-Penrose ’90,and Chrusciel-Galloway ’04 to characterise the positivity ofmass by the behaviour of null geodesics near infinity.(See also Penrose-Sorkin-Woolgar ’93)

Thank you for your attention!